Rigidity for C¹ actions on the interval arising from hyperbolicity I: solvable groups

2013/09/20 by Christian Bonatti, Ignacio Monteverde, Bonatti, C. +5 · 1 citation
Computer Science · Mathematics · #20F16 #22F05 #37C85 #37F15 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1309.5277

openalex publication_date 2013/09/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by C¹ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by C¹ diffeomorphisms of the interval.

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